Which numbers have square roots? Reed College Senior Thesis Presentation Splitting Probabilities of p-adic Polynomials Division of Mathematics and Natural Sciences Spring 2003 Asher Auel Advisor: Joe Buhler
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Which numbers have square roots?
Reed College Senior Thesis Presentation
Splitting Probabilities of p-adic Polynomials
Division of Mathematics and Natural Sciences
Spring 2003
Asher Auel
Advisor: Joe Buhler
Which numbers have square roots?
Which numbers have square roots?
It depends on the rules.
Which numbers have square roots?
It depends on the rules.
For whole numbers
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, . . .
only perfect squares have square roots
0, 1, 4, 9, 16, 25, 36, 49, . . .
Which numbers have square roots?
It depends on the rules.
For real numbers on the number line
0−5.165 . . . −1 12 π 4.12310 . . .
Which numbers have square roots?
It depends on the rules.
For real numbers on the number line
0−5.165 . . . −1 12 π 4.12310 . . .
only non-negative numbers have square roots
0 12 π 4.12310 . . .
in particular, only −1 doesn’t have a square root.
√17 = 4.12301 . . . is irrational!
√17 = 4.12301 . . . is irrational!
[Theodorus] was proving to us a certain thing aboutsquare roots, I mean the side (i.e. root) of a squareof three square units and of five square units, thatthese roots are not commensurable in length with theunit length, and he went on in this way, taking all theseparate cases up to the root of seventeen square units,at which point, for some reason, he stopped.
-Plato, Theaetetus
√17 = 4.12301 . . . is irrational!
But we can get close!
√17 = 4.12301 . . . is irrational!
But we can get close!
x x2 − 174.1 (4.1)2 − 17 = −.19 = −2× 10−1
4.12 (4.12)2 − 17 = −.026 = −3× 10−2
4.123 (4.123)2 − 17 = −.00087 . . . = −9× 10−4
4.1 . . . (4.1 . . .)2 − 17 = ε = u× 10−n
4, 4.1, 4.12, 4.123, 4, 1230 . . . →√
17
√17 = 4.12301 . . . is irrational!
A silly attempt, but why must we choose 10 over 2?
x x2 − 173 32 − 17 = −8 = −1× 23
7 72 − 17 = 32 = 1× 25
23 232 − 17 = 512 = 1× 29
279 2792 − 17 = 77824 = 19× 212
? ?2 − 17 = λ = u× 2n
3, 7, 23, 279, . . . → ?
3 = 1 + 1 · 27 = 1 + 1 · 2 + 1 · 22
23 = 1 + 1 · 2 + 1 · 22 + 0 · 23 + 1 · 24
279 = 1 + 1 · 2 + 1 · 22 + 0 · 23 + 1 · 24 + 1 · 28
...
1111111101111010001111010001001...
↓√17
Which p-adic numbers have square roots?
Which p-adic numbers have square roots?
It depends on p.
Which p-adic numbers have square roots?
It depends on p.
For general p-adic numbers,
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there are multiple square roots missing.
a
b
x +ax+b has a root?2 a,b are 2-adic numbers.
a
b
x +ax+b has which root?2 a,b are 2-adic numbers.
J. P. Serre, 1968
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